20 November 1979
INFORMATION PRCCESSING LETTERS
ON THE SOLUTION OF POLYNOMIAL EQUATIONS USING CONTINUED FRACTIONS
Akiviadis G. AKRITAS
~sl~orrnrotr of Applied MathematicsII. Unillersity
ofAthens, Athens 621, Greece
times of pdynomials,
continued fractions
cc Is wdF krxwn that, 3n the treginning of the 19th
century the mathematicians proved the impossibility
ing algebraieaJly polynomial equations of
gra?wr than four and as a result their attention
d on numerical methods. During this period
r conceived the idea to proceed in two steps;
that is, first to isolate the roots and then to approximate them to any desired degree of accuracy. Ap
proximation is a rather trivial task and will not be
discussed in this paper; moreover, we will be mainly
concerned with the real roots.
Isolation of the real roots of a polynomial equatmn is the process of finding real, disjoint intervals
such that each contains exactly one real roqt and
every root is contained in some interval. In order to
itccomplish this, Sturm’s method is the only one
widely known and used since 1830; since it can be
found in the literature [5] we describe it briefly.
For any given .iquare-free polynomial equation
Pix) = 0, Sturm’s method works as follows: we compute amabsolute upper root bound b, so that all the
KU rnr:rtslie in the interval (-b, b), and then we continuously jubclivide (-b, b) until’in each subinterval
cre is at most one root; that is, Siurm’s method
WCSbisecthn in order to isolate the real roots, (As
w know. with the help of Stxm’s sequence - derivecl
e polynomials P apd P’ - we can easily detere exact number of real roots in any subinterval
where n is the degree of the square-free, integral polynomial equation P(x) = 0, and L(IPI,) the length, in
bits, of the maximum coefficient in absolute value.
The read.er immediately feels that Sturm’s method is
very slow; it has been determined that its slowness is
due to the computation of the Sturm sequence.
During this decade new root isolation procedures
appeared in the literature [ 1, pp. l-81. However,
despite the fact that their theoretical computing time
bounds are better than (1), they all have one thing in
common, namely, they all use b&~&n in order to
isolate real roots.
Recently, in our Pla.D. dissertation [l] we developed our own method for the isolation of the real
roots of a polynomial equation, a method which by
far surpasses all the existing ones in beauty, Gmplicity
and speed; moreover, as we shall see, our method is
the only one with polynomiril computing time bound
which isolates the real roots using continued fractions.
It is based on the following:
1s ~$0 be found in the literature.)
k oi Sturm’s metho is the coefficdes~htrr!w41: !hat is, nl’the calculations are performed
in GE rkg of’ integers, rhe coefficients of the polyrw%als HI the Sturm sequence become too large, and
Theorem (Vincent-Uspensky-Akritas).
Let P(x) = 0
be a polynomial equation of degree n > I, with rational coefficients and without multiple roots, and let
A > 0 be the smallest distance between any two of
182
hence, the round-off errors are not at all negligible,
This drawback was overcome, though, in 1970, when
Sturm’s rnethod was programmed in an algebraic
manipulation system [4] ; moreover, it was shown [4]
that its theoretical computing time bound is
o(n13L(IPl,)3)
,
(1)
Volume 9, number 4
its roots. Let m be the smallest index such that
hl-1
A
~>l
and
20 November 1979
INFORMATION PROCESSING LETTERS
F,_lFmA>l+’
3
(2)
en
arbitrary, positive integer elements a11a2, .... a,
transforms P(x) = 0 into an equation P(f) = 0, which
has at most one sign variation; this transformation
can be also written as
where Fk is the kth member of the Fibonacci sequence
1, 1,2,3,5,8,13,21,
(5)
...
and
where Pk/Qk is the kfh convergent to the continued
fraction
( )
El,,= 1 t J l&-l)
rl
- 1.
(3)
al+-
Then the transformation
1
1
a2 +a3 +
1
:K=al t - t
..
a2
.
(Recall that from the law of convergents we have
‘t _t’
l
am E’
(which is equivalent to the series of successive trans.
formations of the form x = ai + l/&j,i = 1,2, .... m)
presented in the form of a continued fraction with
arbitrary, positive, integral elements al, a2, .... am,
transforms the equation P(x) = 0 in.to the equation
F(g) = 0, which has not more than one sign variation,
in the sequence of its coefficients.
The original form of this theorem (that is, without
specifying the quantity m) is due to Vincent alone
[6], and appeared in 1836. The proof is omitted
since it can be found in the literature [ 1,2]. It shodld
be mentioned that Vincent’s theorem was so totally
forgotten that even such a capital work as the
Enzyclopaedie der mathematischen Wissenschaften
ignores it. The author of this paper discovered it in
Uspensky’s Theory ofEquations [5].
This theorem can be used in order to isolate the
real roots of a polynomial equation. The fact that it
holds only for equations without multiple roots does
not restrict the generality, because in the opposite
case all we have to do is to exlpress P(x) in the form
P = I’& Si, where each of the S{s has only single
roots [S, pp.65-691. Each of these single roots is of
multiplicity i for the polynomial P(x) and thus we
see that our theorem can be applied on the Si’s. So,
in the rest of this discussion it is assumed that P(x) = 0
is without multiple roots.
From the statement of the above theorem we
know that a transformation of the form (4), with
pktl
= aktlpk
Qktl = ak+lQk
t Pk-l
+
,
c-h-1.)
Since the elements al, a2, ‘._,a, are arbitrary there is
obviously an infinite number of transformations of
the form (4). However, with the help of Budan’s
theorem we can easily determine those that are of
interest to us; namely, there is a finite number of
them (equal to the number of the positive roots of
P(x) = 0) which lead to an equation witt. exactly
one sign variation in the sequence of its coefficients.
Suppose that P(t) = 0 is one of these equations; then
from the Cardano-Descartes rule of signs we know
that it has one root in the interval (0, m). If &was
this positive root, then thlz corresponding root ji of
P(x) = 0 could be easily obtained from (5). We only
know though that g lies in the interval (0,00);therefore, substituting &!in (5) once by 0 and once by -3
we obtain for the positive root i its isolating interval,
whose unordered endpoints are P,_ r/Qm._l and
P,/Q,. Xnthis fashion we can isolate all the positive
roots of P(x) = 0. If we subsequently repla,ce x by -x
in the original equation, the negative roots become
positive and hence, they ;oo can be isolated in the
way mentioned above. Thus we see that we have a
procedure for isolating all the real roots of P(x) = 0.
The calculation of the quantities ar, a2, ... . a, for the transformations of the form (4) which lead to
an equation with exactly one sign variation - corstitutes the polynomial real root isolation procedure.
Two methods actually result, Vincent’s and Akritas’,
corresponding to the two different ways iit which the
183
~~~$~~~~e
9. m+tn&r 4
:neurpurarion of the ai’ may be performed.
method btisically consists of computing
by a &es of unit incrementations;
+ I, which corresponds to the substitutrsrvx 6 x + I. This ‘brute force’ Jpproach results in
cthod with an exponential behavior, namely, for
wafueeof the ai’s this method will take a long time
[even years in a computer) in order to isolate the
real roots of an equation. Therefore, Vincent’s methis of little practical importance. Examples of this
roach can be found in Vincent’s paper [6], and in
nsky’s book [S, pp. J29- 1371, The reader should
notice that in the preface of his book Uspensky
s that he himself invented this method. A simple
comparison with Vincent’s paper though makes clear
th.dawhat can be considered a contribution on
~~,~e~sky~spart is only the fact that he used the
Ruffirri- Worncr method [3 1 in order to perform the
~ri~~sf~~rrnat~~~~s
x 6 x + 1, whereas Vincent used
$sylor’s cxpanskn rheorem.Moreover, Uspensky
S~SHMS
its ignore Budan’s theorem and, while computin; a particular ai, he performs, after each transDationx + x + 1, the unnecessary transformation
x +- 1/(x + I ), something which Vincent avoids.
Akritas’ method, on the contrary, is an aesthetic:ltly pleasing interpretation of the VincentI_‘spcnsk;-- Akritas theorem. Basically it consists of
immediarely computing a particular ai as the lower
bound b of the positive roots of a polynomial; that is,
ai +- 5, which corresponds to the substitution
x +- x -bb performed on the particular polynomial
tinder consideration. It is obvious that our method is
independent of how big the values of the ai’s are.
(An unsuccessful treatment of the big values of the
:+‘scan be found in Uspensky [S, p.1361. In this discussion it is assumed that b = Las], where ov,is the
r;mallest positive root.) Since the substitutions
Y+- x f I and x +- x + b can be performed in about
the same time [3 1, we easily see that our method
re&ts in enormous savings of computing time, We
have im@emented our method in a computer algebra
system and have been able to show that its computing
time bound is
184
20 November 1979
[NIXIRMATION PROCESSING LETTERS
Table 1
Degree
Sturm
Akritas
5
10
15
20
--
2.05
33.28
156.40
524.42
0.26
0.48
0.94
2.36
Comparing (1) and (6) we clearly see the superiority
of our method.
In this paper we present just one table (Table 1)
showing the observed times for the methods of Sturm
and Akritas (more examples can be found in [ 11).
All times are in seconds and were obtained by using
the SAC-l computer algebra system on the IBM
S/370 Model 165 computer located at the Triangle
Universities Computation Center, where a subroutine
CCL,OCKis available, which reads the computer
clock. All the coefficients of the polynomials (of
degree 5, 10, 15,20) were nonzero, each ten decimal
digits long and randomly generated.
We see that the most efficient way to isolate the
real roots of a polynomial equation is by using continued fractions. In our Ph.D. thesis we have given all
the necessary hints as to how the above mentioned
theorem may be .used in order to isolate the complex
roo:ts as well. We hope to have more to say on this
sub.ject in the near future.
References
[l]
A.G. Akritas, Vincent’s theorem in algebraic manipula-
tion, Ph.D. thesis, Operations Research Program, North
Carolina State University, Raleigh, NC (1978).
[2 j A.C. Akritas, _4 correction on a theorem by Wspensky,
Bull. Greek Math. Sot. 19 (1978) 278-285.
[3 1 A.G. Akritas and S.D. Danielopoulos, On the complexity
of algorithms for the translation of polynomials, Computing, to appear.
[4] L.E. Heindcl, Integer arithmetic algorithms for poly
nomial real zero determination, J. ACM 18 (1971)
533-548.
1-C1 J.V. Uspensky, Theory of Equations (McGraw-Hill,
New York, 1948).
[fi] M. Vincent, Sur la rEsoIution des 6quations num&iques,
J. Math. Pures Appl. 1 (1836) 341-372.